Math 570: Mathematical Logic
Fall Semester 2004
Prof. Ward Henson
Monday, October 18, 2004
Problem Set 3 – Due in class Monday, October 25, 2004.
(Problem 4 revised 10-20-04)
There are five problems (25 points each) and you should do four of them. To
earn full credit requires a careful writeup of each problem, taking care to justify
everything you claim and to explain your ideas clearly.
3.1. Problem. Let Lbe the first-order language whose non-logical symbols
include a constant symbol c, a unary relation symbol U, and a unary function
symbol f. Let Σ be a set of L-sentences in which fdoes not occur, and suppose
Σ`LUfc. Use the Completeness Theorem to show Σ `L∀xUx.
3.2. Problem. Let Lbe any first-order language and let L0=L(P) be the
language that results from Lby adding a new unary predicate symbol P. Suppose
Σ is a set of L0-sentences and that there exists an L-formula ξ(x) such that
Σ`L0P x ↔ξ(x). Show that for every L0-formula ϕthere exists an L-formula
ψsuch that Σ `L0ϕ↔ψ.
3.3. Problem. Let Lbe the first-order language whose only non-logical sym-
bol is a unary relation symbol P. Consider the L-structure A= (N, E), where E
is the set of even natural numbers. Let Tbe the set of natural numbers divisible
by 3. Use automorphisms of Ato prove that Tis not definable in A, even when
parameters are allowed. That is, show that for any L-formula ϕ(x,y1, . . . , yn)
and any k1, . . . , kn∈N,
T6={m∈N|A|=ϕ(m, k1, . . . , kn)}.
3.4. Problem. Let Lbe the first-order language whose only non-logical sym-
bol is a unary function symbol f. Let Kbe the class of all infinite L-structures
such that fAis a permutation of Awith no finite cycles. Let Σ be the set of all
L-sentences σsuch that A |=σfor all A ∈ K. Show that Σ is complete. ( A
permutation πon a set Ahas “no finite cycles” if there do not exist a∈Aand
n > 0 such that πn(a) = a. This implies that for each a∈A, the map taking k
to πk(a) is an injection from Zinto A.)
3.5. Problem. Let Lbe the first-order language whose only non-logical sym-
bol is a unary predicate symbol P. Let Aand Bbe infinite L-structures. Give
an explicit and clear characterization of the condition Aand Bare elementarily
equivalent. (Do not just repeat the definition; your characterization should be
expressed in terms of definite features of the structures themselves.)